Global existence of a dynamical problem.

Question

Prove that all the solutions to the system $$ \begin{cases} \dot x= e^{-y^2}\sin(x^n+y^n), \\ \dot y= x^n\sin(x^n+y^n), \end{cases} $$ where $n$ is a fixed natural number, are defined on $[0,+\infty)$.

Answer 1

Let $X=(x,y):[0,T)\to\mathbb{R}^2$ be a solution with $X(0)=(x_0,y_0)$. We need to show that $T=\infty$. Note that

$$\tag{1} \left\{ \begin{array}{ccc} x(t)= x_0+\int_0^t e^{-y(s)^2}\sin{(x(s)^n+y(s)^n)}&\mbox{ } \\ y(t)= y_0+\int_0^t x(s)^n\sin{(x(s)^n+y(s)^n)} &\mbox{} \end{array} \right. $$

If $T<\infty$, we conclude from $(1)$ that $|x(t)|<\infty$ for $t\in [0,T)$, which implies also that $|y(t)|<\infty$ for $t\in [0,T)$. Thus, there is a ball $B$ such that the solution $X$ does not leave $B$.

Once the function $(e^{-y^2}\sin{(x^n+y^n)},x^n\sin{(x^n+y^n)} )$ is bounded in $B$, we must conclude that $X(t)\to a\in\mathbb{R}^2$ if $t\to T$, which is an absurd.